Dynamical Systems Approach to Computer Simulation Accuracy and Data Analysis

Sauer0208092研究了非线性动力系统的计算方面，强调了混沌动力学在解释长期模拟结果和来自复杂动力过程的数据方面的作用。模拟方面的工作是对确定性建模中发生的微妙偏差的持续研究，这些偏差对结果有宏观影响。是否有可能对典型的非双曲混沌系统进行长期的计算机模拟，以接近真实的系统行为，这在原则上是一个悬而未决的问题。第二个主要领域是正在进行的关于解释产生非周期性数据的物理和生物实验的工作。从实验室和自然过程中收集的时间序列和尖峰时间数据应用随机建模已经有很长的历史了。研究人员研究了如何使用动态系统技术，包括嵌入理论和棘波序列重构，来扩展实验数据确定性建模的能力。计算机模拟是现代科学的重要组成部分。重要的是要知道，在小的建模误差下，用于模拟的数学模型的解是否能准确地定性和定量地表示被模拟的自然现象，以及在小的浮动指针的情况下，模型的计算机解是否能代表模型的解。这项研究为回答类似于物理相关模型的问题奠定了基础。特别是，研究人员参与分离和量化确定性表示的限制，特别是为了长期建模的目的。该项目的第二个主要关注点是在没有第一原理数学或计算机模型的情况下，对从实验系统和自然中收集的数据进行解释。如果可以从过程中测量到物理相关量的时间痕迹，就存在尝试重建过程的动态行为的技术，具有预测或控制过程的潜力。在物理、化学、工程和生物/医学环境中正在识别复杂的确定性时间序列。研究者在扩展概念基础和开发相关的计算实现方面取得了先前的进展，并正在努力增加他们在科学和工程相关背景下研究复杂系统的能力。

Sauer0208092 The investigator studies computational aspects of nonlineardynamical systems, emphasizing the role of chaotic dynamics inthe interpretation of long-time simulation outcomes and data fromcomplicated dynamical processes. The work on simulation is anongoing study of subtle biases occurring in deterministicmodeling that have macroscopic effects on outcomes. It is anopen question whether it is possible in principle for long-termcomputer simulations of typical nonhyperbolic chaotic systems toapproximately match true system behavior. The second major areais ongoing work on the interpretation of physical and biologicalexperiments that generate aperiodic data. There is a long historyof applying random modeling to time series and spike timing datacollected from laaboratory and natural processes. Theinvestigator syudies ways to use dynamical systems techniques,including embedding theory and spike train reconstruction, toextend the power of deterministic modeling for experimental data. Computer simulation is a critical ingredient of modernscience. It is important to know whether solutions of amathematical model used for simulation can be expected to beaccurate qualitative and quantitative representations of thenatural phenomena being modeled, in the face of small modelingerrors, and whether computer solution of the model can representsolutions of the model, in the face of small floating pointerrors. This reseach builds a foundation for answering questionslike this for physically relevant models. In particular theinvestigator is involved in isolating and quantifying thelimitations of deterministic representations, especially for thepurpose of long-term modeling. The second major focus of theproject is the interpretation of data collected from experimentalsystems and nature when no first-principles mathematical orcomputer model is available. If time traces of physicallyrelevant quantities can be measured from the process, techniquesexist to attempt to reconstruct the dynamical behavior of theprocess, with potential to predict or control the process.Complex deterministic time series are being identified inphysical, chemical, engineering and biological/medical settings.The investigator has made previous progress on expanding theseconceptual foundations and developing related computationalimplementations, and is working to increase their power for thestudy of complex systems in scientific and engineering-relatedcontexts.